Perpendicular Bisector

Perpendicular Bisector Definition

A perpendicular bisector is a line that passes through the midpoint of a line segment at a 90° angle, dividing it into two equal parts. The perpendicular bisector intersects the line segment exactly at its midpoint and forms a right angle with it.

What is a Perpendicular Bisector?

A perpendicular bisector is a line, ray, or line segment that:

• Passes through the midpoint of another line segment
• Makes a 90° angle with that line segment

In the figure below, CD is the perpendicular bisector of line segment AB because:

• CD passes through point M, which is the midpoint of AB
• CD forms a right angle with AB at point M

How to Construct a Perpendicular Bisector?

You can construct a perpendicular bisector using just a compass and a straightedge. Here's how:

Steps to Construct a Perpendicular Bisector:

1. Draw a line segment AB.
2. With the compass point at A, draw an arc that has a radius greater than half the length of AB.
3. Without changing the compass width, place the compass at B and draw another arc that intersects with the first arc at two points.
4. Label these intersection points as C and D.
5. Draw a line through points C and D.

The line CD is the perpendicular bisector of AB.

Properties of Perpendicular Bisector

The perpendicular bisector has several important properties:

1. It divides the line segment into two equal parts (AM = MB).
2. It forms right angles (90°) with the original line segment.
3. Any point on the perpendicular bisector is equidistant from the endpoints of the line segment.
4. The perpendicular bisector is the locus of points equidistant from two fixed points.

Perpendicular Bisector Theorem

The perpendicular bisector theorem states:

Every point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.

If P is any point on the perpendicular bisector of line segment AB, then PA = PB.

Converse of Perpendicular Bisector Theorem

The converse of the perpendicular bisector theorem states:

If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the line segment.

If PA = PB, then point P lies on the perpendicular bisector of AB.

Perpendicular Bisectors of a Triangle

The perpendicular bisector of a triangle refers to the perpendicular bisectors of all three sides of the triangle.

Circumcenter

The point where all three perpendicular bisectors of a triangle meet is called the circumcenter. This point has a special property: it is equidistant from all three vertices of the triangle.

The circumcenter serves as the center of the circumscribed circle of the triangle – the circle that passes through all three vertices of the triangle.

The location of the circumcenter depends on the type of triangle:

• In an acute triangle: The circumcenter lies inside the triangle.
• In a right triangle: The circumcenter lies on the hypotenuse.
• In an obtuse triangle: The circumcenter lies outside the triangle.

Solved Examples

Example 1: Finding Points on a Perpendicular Bisector

Question: If the perpendicular bisector of segment AB passes through point (3, 4), and A is at (1, 2), what are the possible coordinates of point B?

Solution: Step 1: If a point lies on the perpendicular bisector of AB, it is equidistant from A and B. Step 2: Let's call the coordinates of B (x, y). Step 3: The distance from (3, 4) to A is: √[(3-1)² + (4-2)²] = √[4 + 4] = √8 = 2√2 Step 4: The distance from (3, 4) to B must also be 2√2: √[(3-x)² + (4-y)²] = 2√2 Step 5: Solving this equation: (3-x)² + (4-y)² = 8 Step 6: This represents a circle with center (3, 4) and radius 2√2. Step 7: We know that A and B are equidistant from the perpendicular bisector. Step 8: One possible solution is B(5, 6).

Example 2: Constructing a Circumcircle

Question: Construct the circumcircle of triangle PQR with vertices at P(0, 0), Q(4, 0), and R(2, 3).

Solution: Step 1: Draw triangle PQR. Step 2: Construct the perpendicular bisector of side PQ. Step 3: Construct the perpendicular bisector of side QR. Step 4: The intersection of these two perpendicular bisectors gives the circumcenter O. Step 5: With O as center and OP as radius, draw a circle. This is the circumcircle of triangle PQR.

Practice Problems

1. Construct the perpendicular bisector of line segment CD of length 7 cm.
2. If point P lies on the perpendicular bisector of segment MN, and PM = 5 cm, what is PN?
3. The vertices of triangle ABC are A(1, 1), B(5, 1), and C(3, 5). Find the coordinates of its circumcenter.
4. Prove that the perpendicular bisectors of the sides of a rhombus form a rectangle.